#8703067

order	512
nilpotency class	2
automorphism	?
homology	Abelianization (H₁) = C₂⁶ · Schur multiplier (H₂) = C₂¹²

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, commutator, Frattini, socle, cosocle	C₂³	C₂⁶
prime radical	#40914	C₂
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

4361 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 40
transitive degree	μ_t(G) = 256

A minimal faithful permutation representation — 6 generators on 40 points:

(9,10,14,17)(11,16,18,15)(12,21,22,13)(19,20,24,23)(26,31)(27,36)(28,35)(38,39);(1,2)(3,6)(4,7)(5,8)(11,18)(15,16)(19,24)(20,23)(25,26)(27,32)(28,34)(29,35)(30,31)(33,36)(37,39)(38,40);(1,3)(2,5)(4,8)(6,7)(9,11,14,18)(10,15,17,16)(12,19,22,24)(13,20,21,23)(25,27,30,36)(26,32,31,33)(28,37,35,40)(29,38,34,39);(3,8)(5,6)(9,11)(10,16)(12,19)(13,23)(14,18)(15,17)(20,21)(22,24)(25,27)(26,33)(28,37)(29,39)(30,36)(31,32)(34,38)(35,40);(1,2)(3,5)(4,7)(6,8)(9,12,14,22)(10,13,17,21)(11,19,18,24)(15,20,16,23)(25,28)(26,34)(27,37)(29,31)(30,35)(32,39)(33,38)(36,40);(1,3)(2,6)(4,8)(5,7)(9,13)(10,12)(11,20)(14,21)(15,19)(16,24)(17,22)(18,23)(25,29,30,34)(26,35,31,28)(27,38,36,39)(32,40,33,37)

Polycyclic presentation:

2^9|a^2=gh,c^2=h,e^2=gh,f^2=g,b^a=bg,c^a=ch,d^a=dg,e^a=eh,c^b=ci,d^b=dh,d^c=dgi,f^d=fgi,f^e=fgi

Conjugacy classes & character table

89 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 8610263);